Harmonic dipoles and the relaxation of the neo-Hookean energy in 3D elasticity

TL;DR

This paper addresses the challenge of minimizing neo-Hookean energy in 3D elasticity by proposing a relaxed energy framework that excludes cavitation and dipoles, introducing a new explicit energy penalization.

Contribution

It introduces a novel relaxation of the neo-Hookean energy in axisymmetric maps without cavitation, including an explicit penalization for dipole formation, advancing the understanding of energy minimization in elasticity.

Findings

Proposed a relaxation framework for neo-Hookean energy in 3D elasticity.

Developed an explicit energy penalization to prevent dipole formation.

Established a lower bound for the relaxed energy similar to harmonic map energies.

Abstract

We consider the problem of minimizing the neo-Hookean energy in \(3D\). The difficulty of this problem is that the space of maps without cavitation is not compact, as shown by Conti \& De Lellis with a pathological example involving a dipole. In order to rule out this behaviour we consider the relaxation of the neo-Hookean energy in the space of axisymmetric maps without cavitation. We propose a minimization space and a new explicit energy penalizing the creation of dipoles. This new energy, which is a lower bound of the relaxation of the original energy, bears strong similarities with the relaxed energy of Bethuel-Brezis-H\'elein in the context of harmonic maps into the sphere.